**Self-avoiding polymer**

**Simulations: **The dimerization method was used to generate samples of 10^{8} SAWs on a cubic lattice,

for N =16, 32,...,1024.

By measuring the probability that a generated SAW does not intersect the boundary, we obtain

N^{-Δγ}

The phantom polymer results provide an excellent approximation, except for very thin cones.

The green dashed line is from an ε-expansion, treating the surface of the cone as a weak perturbation:

"Apex exponents for polymer-probe interactions,"

M. Slutsky, R. Zandi, Y. Kantor, M. Kardar, PRL

94, 198303 (2005)

**ε-expansion: **A better treatment is to start with impenetrable boundaries,

along the lines of a similar computation for a wedge

"Critical behaviour at an edge," J.L. Cardy, J. Phys. A: Math. Gen. 16 3617

"Polymer-mediated entropic forces between scale-free objects,"

M. F. Maghrebi, Y. Kantor, M. Kardar, PRE

86, 061801 (2012)The loop-correction can only be calculated numerically in general, but for small opening angles we find

Is the exponent for the singular dependence on angle ?

Current simulations are not sufficient to answer this question